Math 7 · Fractions, decimals and percents
Parts of a whole, written three ways
A fraction is a division waiting to be done. Doing it gives a decimal that either stops or repeats forever, and writing it out of 100 gives a percent. Adding and subtracting fractions comes down to one idea: you can only add pieces of the same size.
- 1. The words, first
- 2. Fractions as decimals
- 3. Adding and subtracting fractions
- 4. Percents from 1% to 100%
- 5. What costs marks
The words, first
The idea: Common denominator matters most: it is what makes the pieces the same size.
| Word | What it means |
|---|---|
| Numerator, denominator | The top and bottom of a fraction. The denominator says how many equal pieces the whole is cut into; the numerator says how many you have. |
| Equivalent fractions | Fractions worth the same, like 1/2, 3/6 and 50/100. Multiply or divide the top and bottom by the same number to get one from another. |
| Common denominator | A denominator two fractions can share. For 1/4 and 1/6, 12 works, because 4 and 6 both go into 12. |
| Lowest terms | Top and bottom have no common factor but 1: 3/4, not 9/12. |
| Mixed number | A whole number and a fraction, like 2½. The same amount as an improper fraction, top bigger than bottom, is 5/2. |
| Terminating decimal | A decimal that stops: 3/8 = 0.375. |
| Repeating decimal | A decimal whose digits repeat forever: 2/3 = 0.666… In class the repeating part is often shown with a bar over it. |
| Percent | Per hundred. 35% means 35/100, or 0.35. |
Fractions as decimals
The idea: Divide the numerator by the denominator. Either the division ends, or a remainder comes back and the digits start repeating.
3/8 = 3 ÷ 8 = 0.375 · 5/6 = 5 ÷ 6 = 0.8333… · 5/11 = 0.4545…
Why a decimal repeats. Dividing 5 by 6, every step after the first leaves a remainder of 2, so the next digit is always 3. Dividing by 6, the only possible remainders are 0 to 5, so sooner or later one comes back — unless it is 0, and the decimal stops.
Which fractions stop. In lowest terms, only those whose denominator is made of 2s and 5s multiplied together, such as 4, 8, 20 or 25, because only those can be rewritten over 10, 100 or 1000: 7/20 = 35/100 = 0.35. Simplify first: 3/15 = 1/5 = 0.2 stops, even though 15 has a 3 in it.
Patterns worth knowing. Ninths repeat their numerator: 1/9 = 0.111…, so 0.777… = 7/9. Elevenths repeat a pair that is 9 times the numerator: 1/11 = 0.0909…, 2/11 = 0.1818…, 5/11 = 0.4545…
Back the other way. A terminating decimal is a fraction over 10, 100 or 1000: 0.375 = 375/1000 = 3/8. And a rounded repeating decimal is only close: 0.67 is not 2/3.
Adding and subtracting fractions
The idea: Pieces must be the same size before you can count them together. Find a common denominator, rewrite both fractions, then add or subtract the numerators only.
Why the denominators are not added. 1/2 + 1/3 is not 2/5, which is less than the half you started with. In sixths the pieces match: 3/6 + 2/6 = 5/6. The denominator names the size of the piece, and that does not change when you collect more of them.
3/4 − 1/6 = 9/12 − 2/12 = 7/12
Worked example. A recipe needs 2½ cups of flour for the dough and 1⅔ cups for the topping. How much altogether?
- Common denominator for 2 and 3 is 6: ½ = 3/6 and ⅔ = 4/6.
- Add wholes and fractions: 2 + 1 = 3, and 3/6 + 4/6 = 7/6.
- 7/6 is more than a whole: it is 1⅙, so the total is 4⅙ cups.
- Estimate: 2.5 + 1.7 = 4.2, close to 4⅙ ✓.
When the fraction is too small to subtract. A board is 3⅙ m long and you cut off 1⅔ m. In sixths, that is 3 and 1/6 take away 1 and 4/6, but 1/6 is less than 4/6. Borrow a whole: 3 and 1/6 is the same as 2 and 7/6. Then 2 and 7/6 take away 1 and 4/6 leaves 1 and 3/6, which is 1½ m.
Percents from 1% to 100%
The idea: A percent is a fraction out of 100. To find a percent of a number, change the percent to a decimal and multiply — or build it from 10%.
7/20 = 35/100 = 35% · 3/8 = 0.375 = 37.5% · 35% of 60 = 0.35 × 60 = 21
Fraction to percent. If the denominator goes into 100, scale up: 17 out of 20 on a quiz is 85/100, or 85%. If not, divide, then move the point two places right.
Building from 10%. 10% of a number moves the point one place left. For 35% of 60: 10% is 6, so 30% is 18, and 5% is half of 6, which is 3. Together, 18 + 3 = 21.
Worked example. A hoodie costs $48 and is 25% off. 25% is ¼, so you save 48 ÷ 4 = $12, and you pay 48 − 12 = $36.
What costs marks
The idea: Two are about fractions, two about decimal places, and one about reading the question.
- Adding the denominators. 1/2 + 1/3 = 5/6, not 2/5.
- Subtracting the fractions the easy way round. 3⅙ − 1⅔ is 1½, not 2½. Borrow a whole.
- Treating a rounded decimal as exact. 0.67 is not 2/3.
- Writing 3% as 0.3. It is 0.03: two places, every time.
- Giving the discount when the question asked the price.